In this chapter, you will learn about simplifying algebraic expressions by expanding them. Expanding an algebraic expression allows you to change the form of an expression without changing the output values it gives.

Rewriting an expression in a different form can be useful for simplifying calculations and comparing expressions. We use two main tools to simplify expressions: we combine like terms and/or use the distributive property.

8.1 Expanding algebraic expressions 147

8.2 Simplifying algebraic expressions 152

8.3 Simplifying quotient expressions 155

8.4 Squares, cubes and roots of expressions 160

 

8 Algebraic expressions 2

8.1 Expanding algebraic expressions

multiply often or multiply once: it is your choice

1. (a) Calculate 5 \times 13 and 5 \times 87 and add the two answers.

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(b) Add 13 and 87, and then multiply the answer by 5.

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(c) If you do not get the same answer for questions 1(a) and 1(b), you have made a mistake. Redo your work until you get it right.

The fact that, if you work correctly, you get the same answer for questions 1(a) and 1(b) is an example of a certain property of addition and multiplication called the distributive property. You use this property each time you multiply a number in parts. For example, you may calculate 3 \times 24 by calculating

3 \times 20 and 3 \times 4, and then add the two answers:

3 \times 24 = 3 \times 20 + 3 \times 4

What you saw in question 1 was that 5 \times 100 = 5 \times 13 + 5 \times 87.

This can also be expressed by writing 5(13 + 87).

2. (a) Calculate 10 \times 56.

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(b) Calculate 10 \times 16 + 10 \times 40.

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3. Write down any two numbers smaller than 100. Let us call them x andy. (a) Add your two numbers, and multiply the answer by 6.

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(b) Calculate 6 \times x and 6 \times y and add the two answers.

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(c) If you do not get the same answers for (a) and (b) you have made a mistake somewhere. Correct your work.

4. Complete the table.

(a)	x	1	2	3	4	5
	3(x + 2)	9	12	15	18	21
	3x + 6	9	12	15	18	21
	3x + 2	5	8	11	14	17
	3(x - 2)	-3	0	3	6	9
	3x - 6	-3	0	3	6	9
	3x - 2	1	4	7	10	13

(b) If you do not get the same answers for the expressions 3(x + 2) and 3x + 6, and for 3(x - 2) and 3x - 6, you have made a mistake somewhere. Correct your work.

In algebra we normally write 3(x + 2) instead of 3 \times (x + 2). The expression 3 \times (x + 2) does not mean that you should first multiply by 3 when you evaluate the expression for a certain value of x. The brackets tell you that the first thing you should do is add the value(s) of x to 2 and then multiply the answer by 3.

However, instead of first adding the values within the brackets and then multiplying the answer by 3 we may just do the calculation 3 \times x + 3 \times 2 = 3x + 6 as shown in the table.

(c) Which expressions amongst those given in the table are equivalent? Explain.

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(d) For what value(s) of x is 3(x + 2) = 3x + 2?

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(e) Try to find a value of x such that 3(x + 2) ≠ 3x + 6.

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5. (a) Determine the value of 5x + 15 if x = 6.

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(b) Determine the value of 5(x + 3) if x = 6.

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(c) Can we use the expression 5x + 15 to calculate the value of 5(x + 3) for any values of x? Explain.

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6. Complete the flow diagrams.

(a) (b)

(c) (d)

(e) (f)

7. (a) Which of the above flow diagrams produce the same output numbers?

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(b) Write an algebraic expression for each of the flow diagrams in question 6.

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product expressions AND sum expressions

1. Complete the following:

(a) (3 + 6) + (3 + 6) + (3 + 6) + (3 + 6) + (3 + 6)

=

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\times (

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)

(b) (3 + 6) + (3 + 6) + (3 + 6) + (3 + 6) + (3 + 6)

= (3 + 3 +

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) + (

---

)

= (

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\times

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) + (

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\times

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)

2. Complete the following:

(a) (3x + 6) + (3x + 6) + (3x + 6) + (3x + 6) + (3x + 6)

=

---

(

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)

(b) (3x + 6) + (3x + 6) + (3x + 6) + (3x + 6) + (3x + 6)

= (3x + 3x +

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) + (

---

)

= (

---

\times

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)

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(

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\times

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)

3. In each case, write an expression without brackets that will give the same results as the given expression.

(a) 3(x + 7) (b) 10(2x + 1)

(c) x(4x + 6) (d) 3(2p + q)

(e) t(t + 9) (f) x(y + z)

(g) 2b(b + a - 4) (h) k2(k - m)

4. (a) Complete the table for the given values of x, y and z.

	3(x + 2y + 4z)	3x + 6y + 12z	3x + 2y + 4z
x = 1 y = 2 z = 3	3(1 + 4 + 12) = 3 \times 17 = 51	3 + 12 + 36 = 51	3 + 4 + 12 = 19
x = 10 y = 20 z = 30	3(10 + 40 + 120) = 3 \times 170 = 510	30 + 120 + 360 = 510	30 + 40 + 120 = 190
x = 23 y = 60 z = 100	3(23 + 120 + 400) = 3 \times 543 = 1 629	69 + 360 + 1 200 = 1 629	69 + 120 + 400 = 589
x = 14 y = 0 z = 1	3(14 + 0 + 4) = 3 \times 18 = 54	42 + 0 + 12 = 54	42 + 0 + 4 = 46
x = 5 y = 9 z = 32	3(5 + 18 + 128) = 3 \times 151 = 453	15 + 54 + 384 = 453	15 + 18 + 128 = 161

(b) Which sum expression and product expression are equivalent?

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5. For each expression, write an equivalent expression without brackets.

(a) 2(x2 + x + 1) (b) p(q + r + s)

(c) -3(x + 2y + 3z) (d) x(2x2 + x + 7)

(e) 6x(8 - 2x) (f) 12x(4 - x)

(g) 3x(8x - 5) - 4x(6x - 5) (h) 10x(3x(8x - 5) - 4x(6x - 5))

8.2 Simplifying algebraic expressions

expand, rearrange and then combine like terms

1. Write the shortest possible equivalent expression without brackets.

(a) x + 2(x + 3) (b) 5(4x + 3) + 5x

(c) 5(x + 5) + 3(2x + 1) (d) (5 + x)2

(e) -3(x2 + 2x - 3) + 3(x2 + 4x) (f) x(x - 1) + x + 2

2. (a) Evaluate x(x + 2) + 5x2 - 2x for x = 10.

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(b) Evaluate 6x2 for x = 10.

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(c) Can we use the expression 6x2 to calculate the values of the expression x(x + 2) + 5x2 – 2x for any given value of x? Explain.

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This is how a sum expression for x(x + 2) + 5x2 - 2x can be made:

x(x + 2) + 5x2 - 2x = x \times x + x \times 2 + 5x2 - 2x

= x2 + 2x + 5x2 - 2x

= x2 + 5x2 + 2x - 2x [Rearrange and combine like terms]

= 6x2 + 0

= 6x2

3. Evaluate the following expressions for x = -5:

(a) x + 2(x + 3) (b) 5(4x + 3) + 5x

(c) 5(x + 5) + 3(2x + 1) (d) (5 + x)2

(e) -3(x2 + 2x - 3) + 3(x2 + 4x) (f) x(x - 1) + x + 2

4. Complete the table for the given values of x, y and z.

x	100	80	10	20	30
y	50	40	5	5	20
z	20	30	2	15	10
x + (y - z)	130	90	13	10	40
x - (y - z)	70	70	7	30	20
x - y - z	30	10	3	0	0
x - (y + z)	30	10	3	0	0
x + y - z	130	90	13	10	40
x - y + z	70	70	7	30	20

5. Say whether the following statements are true or false. Refer to the table in question 4.For any values of x, y and z:

(a) x + (y - z) = x + y – z (b) x - (y - z) = x - y - z

6. Write the expressions without brackets. Do not simplify.

(a) 3x - (2y + z) (b) -x + 3(y - 2z)

We can simplify algebraic expressions by using properties of operations as shown:

(5x + 3) - 2(x + 1)

Hence 5x + 3 - 2x - 2

Hence 5x - 2x + 3 - 2

Hence 3x + 1

7. Write an equivalent expression without brackets for each of the following expressions and then simplify:

(a) 22x + (13x - 5) (b) 22x - (13x - 5)

(c) 22x - (13x + 5) (d) 4x - (15 - 6x)

8. Simplify.

(a) 2(x2 + 1) - x - 2 (b) -3(x2 + 2x - 3) + 3x2

8.3 Simplifying quotient expressions

FROM QUOTIENT EXPRESSIONS TO SUM EXPRESSIONS

1. Complete the table for the given values of x.

x	1	7	-3	-10
7x2 + 5x	7 + 5 = 12	(7 \times 49) + (5 \times 7) = 343 + 35 = 378	(7 \times 9) + (5 \times -3) = 63 -15 = 48	700 - 50 = 650
7x + 5	7 + 5 = 12	49 + 5 = 54	-21 + 5 = -16	-70 + 5 = -65
7x + 5x	7 + 5 = 12	49 + 35 = 84	-21 -15 = -36	-70 - 50 = -120
7x2 + 5	7 + 5 = 12	(7 \times 49) + 5 = 343 + 5 = 348	63 + 5 = 68	700 + 5 = 705

2. (a) What is the value of 7x + 5 for x = 0?

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(b) What is the value of for x = 0?

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(c) Which of the two expressions, 7x + 5 or , requires fewer calculations? Explain.

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requires five calculations: three multiplications, one addition and

one division.

(d) Are the expressions 7x + 5 and equivalent, x = 0 excluded? Explain.

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(e) Are there any other expressions that are equivalent to from those given in the table? Explain.

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3. Complete the table for the given values of x.

x	5	10	–5	–10
10x - 5x2	50 - 125 = -75	100 -500 = -400	-50 - 125 = -175	-100 – 500 = -600
5x	5 \times 5 = 25	5 \times 10 = 50	5 \times -5 =-25	5 \times -10 = -50
2 - x	2 - 5 = -3	2 - 10 = -8	2 - (-5) = 7	2 -(-10) = 12

(a) What is the value of 2 - x for x = 0?

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(b) What is the value of for x = 0?

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(c) Are the expressions 2 - x and equivalent, x = 0 excluded? Explain.

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(d) Which of the two expressions 2 - x or requires fewer calculations? Explain.

2 - x. It requires only one calculation. requires six calculations:

four multiplications, one subtraction and one division.

How is it possible that = 7 x + 5 and = 2 - x for all admissible values

of x? We say x = 0 is not an admissible value of x because division by 0 is not allowed.

One of the methods for finding equivalent expressions for algebraic fractions is by means of division:

= (7x2 + 5x) [just as = 3 \times ]

= ( \times 7x2) + ( \times 5x) [distributive property]

= +

= 7x + 5 [provided x ≠ 0]

4. Use the method shown on the previous page to simplify each fraction below.

(a) (b)  

(c) (d)  

5. (a) Evaluate for x = -1.

(b) For the expression to be equivalent to 5 + which value of x must be excluded? Why?

x = 0, because x2 = 0 and division by 0 is undefined and therefore not allowed.

6. Simplify the following expressions:

(a) (b)  

7. Evaluate:

(a) for x = 2 (b) for n = 4

8. Simplify.

(a) (b)  

9. When Natasha and Lebogang were asked to evaluate the expression for x = 10, they did it in different ways.

Natasha's calculation: Lebogang's calculation:

10 + 2 +  

= 12 =  

= 12

Explain how each of them thought about evaluating the given expression.

Natasha first simplified the expression to x + 2 + and then

evaluated the simplified expression. Lebogang simply substituted x = 10

in the original expression.

8.4 Squares, cubes and roots of expressions

simplifying squares and cubes

Study the following example:

(3x)2 = 3x \times 3x

= 3 \times x \times 3 \times x

= 3 \times 3 \times x \times x

= 9x2

1. Simplify the expressions.

(a) (2x)2 (b) (2x2)2 (c) (-3y)2

2. Simplify the expressions.

(a) 25x - 16x (b) 4y + y + 3y (c) a + 17a - 3a

3. Simplify.

(a) (25x - 16x)2 (b) (4y + y + 3y)2 (c) (a + 17a - 3a)2

Study the following example:

(3x)3 = 3x \times 3x \times 3x

= 3 \times x \times 3 \times x \times 3 \times x

= 3 \times 3 \times 3 \times x \times x \times x

= 27x3

4. Simplify the following:

(a) (2x)3 (b) (-x)3

(c) (5a)3 (d) (7y2)3

(e) (-3m)3 (f) (2x3)3

5. Simplify.

(a) 5a - 2a (b) 7x + 3x (c) 4b + b

6. Simplify.

(a) (5a - 2a)3 (b) (7x + 3x)3 (c) (4b + b)3

(d) (13x - 6x)3 (e) (17x + 3x)3 (f) (20y -14y)3

square and cube roots of expressions

1. Thabang and his friend Vuyiswa were asked to simplify .

Thabang reasoned as follows:

To find the square root of a number is the same as asking yourself the question: "Which number was multiplied by itself?" The number that is multiplied by itself is 2a2 and therefore = 2a2

Vuyiswa reasoned as follows:

I should first simplify 2a2 \times 2a2 to get 4a4 and then calculate = 2a2

Which of the two methods do you prefer? Explain why.

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2. Say whether each of the following is true or false. Give a reason for your answer.

(a) = 6x (b) =5x2

3. Simplify.

(a) y6 \times y6 (b) 125x2 + 44x2

4. Simplify.

(a) (b)

(c) (d)  

(e) (f)

5. What does it mean to find the cube root of 8x3 written as ?

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6. Simplify the following:

(a) 2a \times 2a \times2a (b) 10b3 \times 10b3 \times 10b3

(c) 3x3 \times 3x3 \times 3x3 (d) -3x3 \times -3x3 \times -3x3

7. Determine the following:

(a) (b)

(c) (d)

8. Simplify the following expressions:

(a) 6x3 + 2x3 (b) -m3 - 3m3 - 4m3

9. Determine the following:

(a) (b)  

(c) (d)

1. Simplify the following:

(a) 2(3b + 1) + 4 (b) 6 - (2 + 5e)

(c) 18mn + 22mn + 70mn (d) 4pqr + 3 + 9pqr

2. Evaluate each of the following expressions for m = 10:

(a) 3m2 + m + 10 (b) 5(m2 - 5) + m2 + 25

3. (a) Simplify:

= \times 4b + 6 = + = 2b + 3

(b) Evaluate the expression for b = 100.

= 2b + 3 = 200 + 3 = 203

4. Simplify.

(a) (4g)2 (b) (6y)3 (c) (7s + 3s)2

5. Determine the following:

(a) (b) (c)

In this chapter you will solve equations by applying inverse operations. You will also solve equations that contain exponents.

9.1 Thinking forwards and backwards 167

9.2 Solving equations using the additive and multiplicative inverses 170

9.3 Solving equations involving powers 172

 

9 Algebraic equations 2

9.1 Thinking forwards and backwards

doing and undoing what has been done

1. Complete the flow diagram by finding the output values.

 

2. Complete the table.

x	-3	-2	0	5	17
2x	-6	-4	0	10	34

3. Evaluate 4x if:

(a) x = -7 (b) x = 10 (c) x = 0

4. (a) Complete the flow diagram by finding the input values.

 

(b) Puleng put another integer into the flow diagram and got -68 as an answer. Which integer did she put in? Show your calculation.

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(c) Explain how you worked to find the input numbers when you did question (a).

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5. (a) Complete the table.

x	1	3	5	8	18
5x	5	15	25	40	90

(b) Complete the flow diagrams.

(c) Explain how you completed the table.

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6. Look again at questions 1 to 5. For each question, say whether the question required a doing or an undoing process. Give an explanation for your answer (for example: input to output).

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7. (a) Complete the flow diagrams below.

(b) What do you observe?

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8. (a) Complete the flow diagrams below.

(b) What do you observe?

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9. (a) Complete the flow diagrams below.

(b) What do you observe?

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10. (a) Complete the flow diagram below.

 

(b) What calculations will you do to determine what the input number was when the output number is 20?

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Solve the following problems by undoing what was done to get the answer:

11. When a certain number is multiplied by 10 the answer is 150. What is the number?

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12. When a certain number is divided by 5 the answer is 1. What is the number?

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13. When 23 is added to a certain number the answer is 107. What is the original number?

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14. When a certain number is multiplied by 5 and 2 is subtracted from the answer, the final answer is 13. What is the original number?

The number is 3 (because = 3).

9.2 Solving equations using the additive and

multiplicative inverses

finding the unknown

Consider the equation 3x + 2 = 23.

We can represent the equation 3x + 2 = 23 in a flow diagram, where x represents an unknown number:

 

When you reverse the process in the flow diagram, you start with the output number 23, then subtract 2 and then divide the answer by 3:

We can write all of the above reverse process as follows:

Subtract 2 from both sides of the equation:

3x + 2 – 2 = 23 - 2

3x = 21

Divide both sides by 3:

=  

x = 7

We say x = 7 is the solution of 3x + 2 = 23 because 3 \times 7 + 2 = 23. We say that x = 7 makes the equation 3x + 2 = 23 true.

Solve the equations below by using the additive and multiplicative inverses. Check your answers.

1. x + 10 = 0 2. 49x + 2 = 100

3. 2x = 1 4. 20 = 11 - 9x

Example: Solve for x: 7x + 3x = 10

10x = 10

=  

x = 1

5. 4x + 6x = 20 6. 5x = 40 + 3x

7. 3x + 1 - x = 0 8. x + 20 + 4x = -55

 

9.3 Solving equations involving powers

1. Complete the table.

x	1	3	5	7
2x	2	8	32	128

2. Complete the table.

x	0	2	3	5
3x	1	9	27	243

Karina solved the equation 3x = 27 as follows:

3x = 27

Hence 3x = 33

Hence x = 3

3. Now use Karina's method and solve for x in each of the following:

(a) 2x = 32 (b) 4x = 16 (c) 6x = 216 (d) 5x + 1 = 125

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In this chapter, you will learn how to construct, or draw, different lines, angles and shapes. You will use drawing instruments, such as a ruler, to draw straight lines, a protractor to measure and draw angles, and a compass to draw arcs that are a certain distance from a point. Through the various constructions, you will investigate some of the properties of triangles and quadrilaterals; in other words, you will find out more about what is always true about all or certain types of triangles and quadrilaterals.

10.1 Bisecting lines 175

10.2 Constructing perpendicular lines 177

10.3 Bisecting angles 179

10.4 Constructing special angles without a protractor 181

10.5 Constructing triangles 182

10.6 Properties of triangles 185

10.7 Properties of quadrilaterals 187

10.8 Constructing quadrilaterals 189

10 Construction of geometric figures

10.1 Bisecting lines

When we construct, or draw, geometric figures, we often need to bisect lines or angles.Bisect means to cut something into two equal parts. There are different ways to bisect a line segment.

Bisecting a line segment with a ruler

1. Read through the following steps.

Step 1: Draw line segment AB and determine its midpoint.   Step 2: Draw any line segment through the midpoint. The small marks on AF and FB show that AF and FB are equal.

2. Use a ruler to draw and bisect the following line segments: AB = 6 cm and XY = 7 cm.

In Grade 6, you learnt how to use a compass to draw circles, and parts of circles called arcs. We can use arcs to bisect a line segment.

Bisecting a line segment with a compass and ruler

1. Read through the following steps.

Step 1 Place the compass on one endpoint of the line segment (point A). Draw an arc above and below the line. (Notice that all the points on the arc aboveand below the line are the same distance from point A.) Step 2 Without changing the compass width, place the compass on point B. Draw an arc above and below the line so that the arcs cross the first two. (The two points where the arcs cross are the same distance away from point A and from point B.) Step 3 Use a ruler to join the points where the arcs intersect.This line segment (CD) is the bisector of AB. Intersect means to cross or meet. A perpendicular is a line that meets another line at an angle of 90°.

2. Work in your exercise book. Use a compass and a ruler to practise drawing perpendicular bisectors on line segments.

10.2 Constructing perpendicular lines

A perpendicular line from a given point

1. Read through the following steps.

Step 1 Place your compass on the given point (point P). Draw an arc across the line on each side of the given point. Do not adjust the compass width when drawing the second arc.	Step 2 From each arc on the line, draw another arc on the opposite side of the line from the given point (P). The two new arcs will intersect.
Step 3 Use your ruler to join the given point (P) to the point where the arcs intersect (Q).	PQ is perpendicular to AB. We also write it like this: PQ ⊥ AB.

2. Use your compass and ruler to draw a perpendicular line from each given point to the line segment:

A perpendicular line at a given point on a line

1. Read through the following steps.

Step 1 Place your compass on the given point (P). Draw an arc across the line on each side of the given point. Do not adjust the compass width when drawing the second arc.	Step 2 Open your compass so that it is wider than the distance from one of the arcs to the point P. Place the compass on each arc and draw an arc above or below the point P. The two new arcs will intersect.
Step 3 Use your ruler to join the given point (P) and the point where the arcs intersect (Q). PQ ⊥ AB

2. Use your compass and ruler to draw a perpendicular at the given point on each line:

 

10.3 Bisecting angles

Angles are formed when any two lines meet. We use degrees (°) to measure angles.

Measuring and classifying angles

In the figures below, each angle has a number from 1 to 9.

1. Use a protractor to measure the sizes of all the angles in each figure. Write your answers on each figure.

(a) (b)

 

2. Use your answers to fill in the angle sizes below.

=

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°

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=

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°

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+ =

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°

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+ =

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°

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+ =

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°

---

+ + =

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°

---

+ =

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°

---

+ + =

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°

---

+ =

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°

---

+ =

---

°

---

+ + =

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°

---

+ + + =

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°

---

+ + + =

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°

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+ + + + =

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°

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3. Next to each answer above, write down what type of angle it is, namely acute, obtuse, right, straight, reflex or a revolution.

Bisecting angles without a protractor

1. Read through the following steps.

Step 1 Place the compass on the vertex of the angle (point B). Draw an arc across each arm of the angle.	Step 2 Place the compass on the point where one arc crosses an arm and draw an arc inside the angle. Without changing the compass width, repeat for the other arm so that the two arcs cross.
Step 3 Use a ruler to join the vertex to the point where the arcs intersect (D). DB is the bisector of A C.

2. Use your compass and ruler to bisect the angles below.

10.4 Constructing special angles without a protractor

constructing angles of 60°, 30° and 120°

1. Read through the following steps.

Step 1 Draw a line segment (JK). With the compass on point J, draw an arc across JK and up over above point J.	Step 2 Without changing the compass width, move the compass to the point where the arc crosses JK, and draw an arc that crosses the first one.
Step 3 Join point J to the point where the two arcs meet (point P).P K = 60°

2. (a) Construct an angle of 60° at point B on the next page.

(b) Bisect the angle you constructed.

(c) Do you notice that the bisected angle consists of two 30° angles?

(d) Extend line segment BC to A. Then measure the angle adjacent to the 60° angle.

What is its size?

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(e) The 60° angle and its adjacent angle add up to

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Constructing angles of 90° and 45°

1. Construct an angle of 90° at point A. Go back to section 10.2 if you need help.

2. Bisect the 90° angle, to create an angle of 45°. Go back to section 10.3 if you need help.

 

10.5 Constructing triangles

In this section, you will learn how to construct triangles. You will need a pencil, a protractor, a ruler and a compass.

A triangle has three sides and three angles. We can construct a triangle when we know some of its measurements, that is, its sides, its angles, or some of its sides and angles.

Constructing triangles

Constructing triangles when three sides are given

1. Read through the following steps. They describe how to construct \triangle}ABC with side lengths of 3 cm, 5 cm and 7 cm.

Step 1 Draw one side of the triangle using a ruler. It is often easier to start with the longest side.	Step 2 Set the compass width to 5 cm. Draw an arc 5 cm away from point A. The third vertex of the triangle will be somewhere along this arc.
Step 3 Set the compass width to 3 cm. Draw an arc from point B. Note where this arc crosses the first arc. This will be the third vertex of the triangle.	Step 4 Use your ruler to join points A and B to the point where the arcs intersect (C).

2. Work in your exercise book. Follow the steps above to construct the following triangles:

(a) \triangle}ABC with sides 6 cm, 7 cm and 4 cm

(b) \triangle}KLM with sides 10 cm, 5 cm and 8 cm

(c) \triangle}PQR with sides 5 cm, 9 cm and 11 cm

Constructing triangles when certain angles and sides are given

3. Use the rough sketches in (a) to (c) below to construct accurate triangles, using a ruler, compass and protractor. Do the construction next to each rough sketch.

• • The dotted lines show where you have to use a compass to measure the length of a side.
• • Use a protractor to measure the size of the given angles.

(a) Construct \triangle}ABC, with two angles andone side given.

 

(b) Construct a \triangle}KLM, with two sides andan angle given.

 

(c) Construct right-angled \triangle}PQR, with thehypotenuse and one other side given.

 

4. Measure the missing angles and sides of each triangle in 3(a) to (c) on the previous page. Write the measurements at your completed constructions.

5. Compare each of your constructed triangles in 3(a) to (c) with a classmate's triangles. Are the triangles exactly the same?

 

10.6 Properties of triangles

The angles of a triangle can be the same size or different sizes. The sides of a triangle can be the same length or different lengths.

properties of equilateral triangles

1. (a) Construct \triangle}ABC next to its rough sketch below.

(b) Measure and label the sizes of all its sides and angles.

 

2. Measure and write down the sizes of the sides and angles of \triangle}DEF on the right.

3. Both triangles in questions 1 and 2 are called equilateral triangles. Discuss with a classmate if the following is true for an equilateral triangle:

• • All the sides are equal.
• • All the angles are equal to 60°.

Properties of isosceles triangles

1. (a) Construct \triangle}DEF with EF = 7 cm, = 50° and = 50°.

Also construct \triangle}JKL with JK = 6 cm, KL = 6 cm and = 70°.

(b) Measure and label all the sides and angles of each triangle.

2. Both triangles above are called isosceles triangles. Discuss with a classmate whether the following is true for an isosceles triangle:

• • Only two sides are equal.
• • Only two angles are equal.
• • The two equal angles are opposite the two equal sides.

The sum of the angles in a triangle

1. Look at your constructed triangles \triangle}ABC, \triangle}DEF and \triangle}JKL above and on the previous page. What is the sum of the three angles each time?

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2. Did you find that the sum of the interior angles of each triangle is 180°? Do the following to check if this is true for other triangles.

(a) On a clean sheet of paper, construct any triangle. Label the angles A, B and C and cut out the triangle.

(b) Neatly tear the angles off the triangle and fit them next to one another.

(c) Notice that , and form a straight angle. Complete: + + =

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°

10.7 Properties of quadrilaterals

A quadrilateral is any closed shape with four straight sides. We classify quadrilaterals according to their sides and angles. We note which sides are parallel, perpendicular or equal. We also note which angles are equal.

Properties of quadrilaterals

1. Measure and write down the sizes of all the angles and the lengths of all the sides of each quadrilateral below.

Square	Rectangle
Parallelogram	Rhombus
Trapezium	Kite

2. Use your answers in question 1. Place a ✓ in the correct box below to show which property is correct for each shape.

Properties	Parallelogram	Rectangle	Rhombus	Square	Kite	Trapezium
Only one pair of sides are parallel						✓
Opposite sides are parallel	✓	✓	✓	✓
Opposite sides are equal	✓	✓	✓	✓
All sides are equal			✓	✓
Two pairs of adjacent sides are equal			✓	✓	✓
Opposite angles are equal	✓	✓	✓	✓
All angles are equal		✓		✓

Sum of the angles in a quadrilateral

1. Add up the four angles of each quadrilateral on the previous page. What do you notice about the sum of the angles of each quadrilateral?

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2. Did you find that the sum of the interior angles of each quadrilateral equals 360°? Do the following to check if this is true for other quadrilaterals.

(a) On a clean sheet of paper, use a ruler to construct any quadrilateral.

(b) Label the angles A, B, C and D. Cut out the quadrilateral.

(c) Neatly tear the angles off the quadrilateral and fit them next to one another.

(d) What do you notice?

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10.8 Constructing quadrilaterals

You learnt how to construct perpendicular lines in section 10.2. If you know how to construct parallel lines, you should be able to construct any quadrilateral accurately.

Constructing parallel lines to draw quadrilaterals

1. Read through the following steps.

Step 1 From line segment AB, mark a point D. This point D will be on the line that will be parallel to AB. Draw a line from A through D.	Step 2 Draw an arc from A that crosses AD and AB. Keep the same compass width and draw an arc from point D as shown.
Step 3 Set the compass width to the distance between the two points where the first arc crosses AD and AB. From the point where the second arc crosses AD, draw a third arc to cross the second arc.	Step 4 Draw a line from D through the point where the two arcs meet. DC is parallel to AB.

2. Practise drawing a parallelogram, square and rhombus in your exercise book.

3. Use a protractor to try to draw quadrilaterals with at least one set of parallel lines.

1. Do the following construction in your exercise book.

(a) Use a compass and ruler to construct equilateral \triangle}ABC with sides 9 cm.

(b) Without using a protractor, bisect .Let the bisector intersect AC at point D.

(c) Use a protractor to measure A B. Write the measurement on the drawing.

2. Name the following types of triangles and quadrilaterals.

A B C

D E F

3. Which of the following quadrilaterals matches each description below? (There may be more than one answer for each.)

parallelogram; rectangle; rhombus; square; kite; trapezium

(a) All sides are equal and all angles are equal.

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(b) Two pairs of adjacent sides are equal.

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(c) One pair of sides is parallel.

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(d) Opposite sides are parallel.

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(e) Opposite sides are parallel and all angles are equal.

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(f) All sides are equal.

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Challenge

1. Construct these triangles:

(a) \triangle}STU, with three angles given: = 45°, = 70° and = 65°.

(b) \triangle}XYZ, with two sides and the angle opposite one of the sides given: = 50°, XY = 8 cm and XZ = 7 cm.

2. Can you find more than one solution for each triangle above? Explain your findings to a classmate.

In this chapter, you will learn more about different kinds of triangles and quadrilaterals, and their properties. You will explore shapes that are congruent and shapes that are similar. You will also use your knowledge of the properties of 2D shapes in order to solve geometric problems.

11.1 Types of triangles 193

11.2 Unknown angles and sides of triangles 195

11.3 Types of quadrilaterals and their properties 200

11.4 Unknown angles and sides of quadrilaterals 204

11.5 Congruency 205

11.6 Similarity 207

11 Geometry of 2D shapes

11.1 Types of triangles

By now, you know that a triangle is a closed 2D shape with three straight sides. We can classify or name different types of triangles according to the lengths of their sides and according to the sizes of their angles.

Naming triangles according to their sides

1. Match the name of each type of triangle with its correct description.

Name of triangle	Description of triangle
Isosceles triangle	All the sides of a triangle are equal.
Scalene triangle	None of the sides of a triangle are equal.
Equilateral triangle	Two sides of a triangle are equal.

2. Name each type of triangle by looking at its sides.

 

Naming triangles according to their angles

Remember the following types of angles:

 

Acute angle Right angle Obtuse angle

(< 90°) (= 90°) (between 90° and 180°)

Study the following triangles; then answer the questions:

 

Acute triangle Right-angled triangle Obtuse triangle

1. Are all the angles of a triangle always equal?

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2. When a triangle has an obtuse angle, it is called an

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triangle.

3. When a triangle has only acute angles, it is called an

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triangle.

4. When a triangle has an angle equal to

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, it is called a right-angled triangle.

investigating the angles and sides of triangles

1. (a) What is the sum of the interior angles of a triangle?

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(b) Can a triangle have two right angles?

Explain your answer.

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(c) Can a triangle have more than one obtuse angle? Explain your answer.

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2. Look at the triangles below. The arcs show which angles are equal.

Equilateral triangle Isosceles triangle Right-angled triangle

(a) \triangle}ABC is an equilateral triangle. What do you notice about its angles?

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(b) \triangle}FEM is an isosceles triangle. What do you notice about its angles?

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(c) \triangle}JKL is a right-angled triangle. Is its longest side opposite the 90° angle?

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(d) Construct any three right-angled triangles on a sheet of paper. Is the longest side always opposite the 90° angle?

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11.2 Unknown angles and sides of triangles

You can use what you know about triangles to obtain other information. When you work out new information, you must always give reasons for the statements you make.

Look at the examples below of working out unknown angles and sides when certain information is given. The reason for each statement is written in square brackets.

 

= = = 60° [Angles in an equilateral \triangle} = 60°]

DE = DF [Given]

= [Angles opposite the equal sides of an isosceles \triangle} are equal]

= 55° [The sum of the interior angles of a \triangle} = 180°; so = 180° – 40° – 85°]

You can shorten the following reasons in the ways shown:

• • Sum of interior angles (∠s) of a triangle (\triangle}) = 180°: Interior ∠s of \triangle}
• • Isosceles triangle has 2 sides and 2 angles equal: Isosceles \triangle}
• • Equilateral triangle has 3 sides and 3 angles equal: Equilateral \triangle}
• • Angles forming a straight line = 180°: Straight line

Working out unknown angles and sides

Find the sizes of unknown angles and sides in the following triangles. Always give reasons for every statement.

1. What is the size of ?	+ + = [Interior ∠s of a \triangle}] 50° + + = 145° + = = – 145° =
2. Determine the size of .	+ + = 180° [Interior ∠s of a \triangle}] 45° + + 60° = 180° = 180° - 45° - 60° = 75°
3. (a) What is the length of KM? (b) Find the size of .	(a) KM = KL = 50 mm (b) = = 38° [Isosceles \triangle}] + + = 180° [Interior ∠s of a \triangle}] + 38° + 38° = 180° = 180° - 38° - 38° = 104°
4. What is the size of ?	+ + = 180° [Interior ∠s of a \triangle}] + 90° + 32° = 180° = 180° - 90°- 32° = 58°

5. (a) Find CB. (b) Find if = 50°.	(a) CB = CA = 8 cm [Isosceles \triangle}] (b) = = 50° + + = 180° [Interior ∠s of a \triangle}] + 50° + 50° = 180° = 180° - 50° - 50° = 80°
6. (a) Find DF. (b) Find if = 100°.	(a) DF = DE = 4 mm [Isosceles \triangle}] (b) + + = 180° [Interior ∠s of a \triangle}] 100° + + = 180° 2 = 180° - 100° = 80° \div 2 = 40°

working out more unknown angles and sides

1. Calculate the size of and .

= [Isosceles \triangle}] 24° + + = 180° [Interior ∠s of a \triangle}] + = 180° - 24° 2 = 156° = 78° = 78°

2. Calculate the size of x.	= = x [Isosceles \triangle}] 80° + x + x = 180° [Interior ∠s of a \triangle}] 2x = 180° - 80° 2x = 100° x = 50°
3. KLM is a straight line. Calculate the size of x and y.
	100° + 50° + x = 180° [Interior ∠s of a \triangle}] x = 180° - 100° - 50° x = 30° 30° + y = 180° [Straight line] y = 180° - 30° y = 150°
4. Angle b and an angle with size 130° form a straight angle. Calculate the size of a and b.
	130° + b = 180° [straight line] b = 180° - 130° b = 50° 130° + 30° + a = 180° [Interior ∠s of a \triangle}] a = 180° - 130° - 30° a = 20°
5. m and n form a straight angle. Calculate the size of m and n.
	m = 60° [equilateral \triangle}] m + n = 180° [straight line] n = 180° - 60° n = 120°

6. BCD is a straight line segment. Calculate the size of x.
	= = x [equilateral \triangle}] A B = 180° - 68° [straight line] A B = 112° 112° + x + x = 180° 2x = 180° - 112° 2x = 68° x = 34°
7. Calculate the size of x and then the size of .
	x + (2x + 40°) + (x + 20°) = 180° [Interior ∠s of a \triangle}] 4x = 180° - 40° - 20° 4x = 120° x = 30° = 2x + 40° = 2(30°) + 40° = 100°
8. Calculate the size of .
	(x + 10°) + (2x - 30°) + (2x - 50°) = 180° [Interior ∠s of a \triangle}] 5x = 180° + 30° + 50° - 10° 5x = 250° x = 50° = 2(50°) - 30° = 70°
9. DNP is a straight line. Calculate the size of x and of y.
	NP = M N = 56° [Isosceles \triangle}] x + 56° + 56° = 180° [Interior ∠s of a \triangle}] x = 68° M D = 180° - 68° [Straight line] M D = 112° N D = NM = y [Isosceles \triangle}] y + y + 112° = 180° [Interior ∠s of a \triangle}] y = 34°

11.3 Types of quadrilaterals and their properties

A quadrilateral is a figure with four straight sides which meet at four vertices. In many quadrilaterals all the sides are of different lengths and all the angles are of different sizes.

You have previously worked with these types of quadrilaterals, in which some sides have the same lengths, and some angles may be of the same size.

the properties of different types of quadrilaterals

1. In each question below, different examples of a certain type of quadrilateral are given. In each case identify which kind of quadrilateral it is. Describe the properties of each type by making statements about the lengths and directions of the sides and the sizes of the angles of each type. You may have to take some measurements to be able to do this.

Question 1(a)

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Question 1(b)

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Question 1(c)

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Question 1(d)

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Question 1(e)

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Question 1(f)

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2. Use your completed lists and the drawings in question 1 to determine if the following statements are true (T) or false (F).

(a) A rectangle is a parallelogram.

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(b) A square is a parallelogram.

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(c) A rhombus is a parallelogram.

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(d) A kite is a parallelogram.

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(e) A trapezium is a parallelogram.

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(f) A square is a rhombus.

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(g) A square is a rectangle.

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(h) A square is a kite.

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(i) A rhombus is a kite.

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(j) A rectangle is a rhombus.

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(k) A rectangle is a square.

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3. Here are some conventional definitions of quadrilaterals:

• • A parallelogram is a quadrilateral with two opposite sides parallel.
• • A rectangle is a parallelogram that has all four angles equal to 90°.
• • A rhombus is a parallelogram with all four sides equal.
• • A square is a rectangle with all four sides equal.
• • Atrapezium is a quadrilateral with one pair of opposite sides parallel.
• • A kite is a quadrilateral with two pairs of adjacent sides equal.

Write down other definitions that work for these quadrilaterals.

(a) Rectangle:

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(b) Square:

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(c) Rhombus:

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(d) Kite:

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(e) Trapezium:

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11.4 Unknown angles and sides of quadrilaterals

finding unknown angles and sides

Find the length of all the unknown sides and angles in the following quadrilaterals. Give reasons to justify your statements. (Also recall that the sum of the angles of a quadrilateral is 360°.)

1.
2.
3. ABCD is a kite.
4. The perimeter of RSTU is 23 cm.

5. PQRS is a rectangle and has a perimeter of 40 cm.

 

11.5 Congruency

what is congruency?

1. \triangle}ABC is reflected in the vertical line (mirror) to give \triangle}KLM.

Are the sizes and shapes of the two triangles exactly the same?

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2. \triangle}MON is rotated 90° around point F to give you \triangle}TUE.

Are the sizes and shapes of \triangle}MON and \triangle}TUE exactly the same?

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3. Quadrilateral ABCD is translated 6 units to the right and 1 unit down to give quadrilateral XRZY.

Are ABCD and XRZY exactly the same?

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In the previous activity, each of the figures was transformed (reflected, rotated or translated) to produce a second figure. The second figure in each pair has the same angles, side lengths, size and area as the first figure. The second figure is thus an accurate copy of the first figure.

Notation of congruent figures

When we name shapes that are congruent, we name them so that the matching, or corresponding, angles are in the same order. For example, in \triangle}ABC and\triangle}KLM on the previous page:

is congruent to (matches and is equal to) . is congruent to . is congruent to .

We therefore use this notation: \triangle}ABC≡\triangle}KML.

Similarly for the other pairs of figures on the previous page:\triangle}MON≡\triangle}ETU and ABCD≡XRZY.

The notation of congruent figures also shows which sides of the two figures correspond and are equal. For example, \triangle}ABC ≡ \triangle}KML shows that:

AB = KM, BC = ML and AC = KL

The incorrect notation \triangle}ABC ≡ \triangle}KLM will show the following incorrect information: = , = ,AB = KL, and AC = KM.

identifying congruent angles and sides

Write down which angles and sides are equal between each pair of congruent figures.

1. \triangle}PQR ≡ \triangle}UCT	2. \triangle}KLM ≡ \triangle}UWC
3. \triangle}GHI ≡ \triangle}QRT	4. \triangle}KJL ≡ \triangle}POQ

11.6 Similarity

In Grade 7, you learnt that two figures are similar when they have the same shape (their angles are equal) but they may be different sizes. The sides of one figure are proportionally longer or shorter than the sides of the other figure; that is, the length of each side is multiplied or divided by the same number. We say that one figure is an enlargement or a reduction of the other figure.

checking for similarity

1. Look at the rectangles below and answer the questions that follow.

 

(a) Look at rectangle 1 and ABCD:

How many times is FH longer than BC?

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How many times is GF longer than AB?

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(b) Look at rectangle 2 and ABCD:

How many times is IL longer than BC?

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How many times is LM longer than CD?

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(c) Is rectangle 1 or rectangle 2 an enlargement of rectangle ABCD? Explain your answer.

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2. Look at the triangles below and answer the questions that follow.

 

(a) How many times is:

• • FG longer than BC?

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  HF longer than AB?

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• • HG longer than AC?

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  IK shorter than BC?

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• • JI shorter than AB?

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  JK shorter than AC?

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(b) Is \triangle}HFG an enlargement of \triangle}ABC? Explain your answer.

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(c) Is \triangle}JIK a reduction of \triangle}ABC? Explain your answer.

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In the previous activity, rectangle KILM is an enlargement of rectangle ABCD. Therefore, ABCD is similar to KILM. The symbol for ‘is similar to' is: ///. So we write: ABCD /// KILM.

The triangles on the previous page are also similar. \triangle}HFG is an enlargement of \triangle}ABC and \triangle}JIK is a reduction of \triangle}ABC.

In \triangle}ABC and \triangle}HFG, = , = and = . We therefore write it like this: \triangle}ABC /// \triangle}HFG.

In the same way, \triangle}ABC /// \triangle}JIK.

using properties of similar and congruent figures

1. Are the triangles in each pair similar or congruent? Give a reason for each answer.

 

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2. Is \triangle}RTU /// \triangle}EFG? Give a reason for your answer.

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3. \triangle}PQR /// \triangle}XYZ. Determine the length of XZ and XY.

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4. Are the following statements true or false? Explain your answers.

(a) Figures that are congruent are similar.

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(b) Figures that are similar are congruent.

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(c) All rectangles are similar.

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(d) All squares are similar.

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1. Study the triangles below and answer the following questions:

(a) Tick the correct answer. \triangle}ABC is:

acute and equilateral

obtuse and scalene

acute and isosceles

right-angled and isosceles.

(b) If AB = 40 mm, what is the length of AC?

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(c) If = 80°, what is the size of and of ?

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(d) \triangle}ABC ≡ \triangle}FDE. Name all the sides in the two triangles that are equal to AB.

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(e) Name the side that is equal to DE.

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(f) If is 40°, what is the size of ?

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2. Look at figures JKLM and PQRS. (Give reasons for your answers below.)

 

(a) What type of quadrilateral is JKLM?

(b) Is JKLM /// PQRS?

(c) What is the size of ?

(d) What is the size of ?

= = 65° [Corresponding angles of similar figures].

(e) What is the length of KL?

When you enlarge or reduce a polygon, you need to enlarge or reduce all its sides proportionally, or by the same ratio. This means that you multiply or divide each length by the same number.

In this chapter, you will explore the relationships between pairs of angles that are created when straight lines intersect (meet or cross). You will examine the pairs of angles that are formed by perpendicular lines, by any two intersecting lines, and by a third line that cuts two parallel lines. You will come to understand what is meant by vertically opposite angles, corresponding angles, alternate angles and co-interior angles. You will be able to identify different angle pairs, and then use your knowledge to help you work out unknown angles in geometric figures.

12.1 Angles on a straight line 213

12.2 Vertically opposite angles 216

12.3 Lines intersected by a transversal 219

12.4 Parallel lines intersected by a transversal 222

12.5 Finding unknown angles on parallel lines 224

12.6 Solving more geometric problems 227

[to come]

12 Geometry of straight lines

12.1 Angles on a straight line

Sum of angles on a straight line

In the figures below, each angle is given a label from 1 to 5.

1. Use a protractor to measure the sizes of all the angles in each figure. Write your answers on each figure.

A

B

2. Use your answers to fill in the angle sizes below.

(a) + =

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° (b) + + =

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°

Finding unknown angles on straight lines

Work out the sizes of the unknown angles below. Build an equation each time as you solve these geometric problems. Always give a reason for every statement you make.

1. Calculate the size of a.	a + 63° = [∠s on a straight line] a = - 63° =
2. Calculate the size of x.	x + 29° + 90° = 180° [∠s on a straight line]
3. Calculate the size of y.

Finding more unknown angles on straight lines

1. Calculate the size of: (a) x (b) E B
2. Calculate the size of: (a) m (b) S R
3. Calculate the size of: (a) x (b) H F
4. Calculate the size of: (a) k (b) T P

5. Calculate the size of: (a) p (b) J R

 

12.2 Vertically opposite angles

What are vertically opposite angles?

1. Use a protractor to measure the sizes of all the angles in the figure. Write your answers on the figure.

2. Notice which angles are equal and how these equal angles are formed.

 

Finding unknown angles

Calculate the sizes of the unknown angles in the following figures. Always give a reason for every statement you make.

1. Calculate x, y and z.	x = ° [vert. opp. ∠s] y + 105° = ° [∠s on a straight line] y = - 105° = z = [vert. opp. ∠s]
2. Calculate j, k and l.
3. Calculate a, b, c and d.

equations using vertically opposite angles

Vertically opposite angles are always equal. We can use this property to build an equation. Then we solve the equation to find the value of the unknown variable.

1. Calculate the value of m.	m + 20° = 100° [vert. opp. ∠s] m = 100° - 20° =
2. Calculate the value of t.
3. Calculate the value of p.
4. Calculate the value of z.

5. Calculate the value of y.
6. Calculate the value of r.

 

12.3 Lines intersected by a transversal

pairs of angles formed by a transversal

A transversal is a line that crosses at least two other lines.

 

When a transversal intersects two lines, we can compare the sets of angles on the two lines by looking at their positions.

The angles that lie on the same side of the transversal and are in matching positions are called corresponding angles (corr.∠s). In the figure, these are corresponding angles:

• • a and e
• • b and f
• • d andh
• • c and g.

1. In the figure, a and e are both left of the transversal and above a line.

Write down the location of the following corresponding angles. The first one is done for you.

b and f:

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d and h:

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c and g:

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Alternate angles (alt.∠s) lie on opposite sides of the transversal, but are not adjacent or vertically opposite. When the alternate angles lie between the two lines, they are called alternate interior angles. In the figure, these are alternate interior angles:

• • d and f
• • c and e.

When the alternate angles lie outside of the two lines, they are called alternate exterior angles. In the figure, these are alternate exterior angles:

• • a and g
• • b and h.

2. Write down the location of the following alternate angles:

d and f:

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c and e:

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a and g:

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b and h:

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Co-interior angles (co-int.∠s) lie on the same side of the transversal and between the two lines. In the figure, these are co-interior angles:

• • c and f
• • d and e.

3. Write down the location of the following co-interior angles:

d and e:

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c and f:

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identifying types of angles

Two lines are intersected by a transversal as shown below.

 

Write down the following pairs of angles:

1. two pairs of corresponding angles:

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2. two pairs of alternate interior angles:

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3. two pairs of alternate exterior angles:

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4. two pairs of co-interior angles:

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5. two pairs of vertically opposite angles:

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12.4 Parallel lines intersected by a transversal

investigating angle sizes

In the figure below left, EF is a transversal to AB and CD. In the figure below right, PQ is a transversal to parallel lines JK and LM.

 

1. Use a protractor to measure the sizes of all the angles in each figure. Write the measurements on the figures.

2. Use your measurements to complete the following table.

Angles	When two lines are not parallel	When two lines are parallel
Corr. ∠s	= ; = = ; = = ; = = ; =	= ; = = ; = = ; = = ; =
Alt. int. ∠s	= ; = = ; =	= ; = = ; =
Alt. ext. ∠s	= ; = = ; =	= ; = = ; =
Co-int. ∠s	+ = + =	+ = + =

3. Look at your completed table in question 2. What do you notice about the angles formed when a transversal intersects parallel lines?

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identifying angles on parallel lines

1. Fill in the corresponding angles to those given.

 

2. Fill in the alternate exterior angles.

 

3. (a) Fill in the alternate interior angles.

(b) Circle the two pairs of co-interior angles in each figure.

4. (a) Without measuring, fill in all the angles in the following figures that are equal tox and y.

(b) Explain your reasons for each x and y that you filled in to your partner.

A B

5. Give the value of x and y below.

12.5 Finding unknown angles on parallel lines

working out unknown angles

Work out the sizes of the unknown angles. Give reasons for your answers. (The first one has been done as an example.)

1. Find the sizes of x, y andz.

x = 74° [alt. ∠ with given 74°; AB // CD] y = 74° [corr. ∠ with x; AB // CD] ory = 74° [vert. opp. ∠ with given 74°] z = 106° [co-int. ∠ with x; AB // CD] orz = 106° [∠s on a straight line]

2. Work out the sizes of p, q and r.
3. Find the sizes of a, b, c and d.
4. Find the sizes of all the angles in this figure. J K
5. Find the sizes of all the angles. (Can you see two transversals and two sets of parallel lines?) C A D B

EXTENSION

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Two angles in the following diagram are given as x and y. Fill in all the angles that are equal to x and y.

sum of the angles in a quadrilateral

The diagram below is a section of the previous diagram.

 

1. What kind of quadrilateral is in the diagram? Give a reason for your answer.

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2. Look at the top left intersection. Complete the following equation:

Angles around a point = 360°

∴ x + y +

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+

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= 360°

3. Look at the interior angles of the quadrilateral. Complete the following equations:

Sum of angles in the quadrilateral = x + y +

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+

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From question 2: x + y +

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+

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= 360°

∴ Sum of angles in a quadrilateral =

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°

12.6 Solving more geometric problems

angle relationships on parallel lines

1. Calculate the sizes of to .
2. Calculate the sizes of x, y and z.
3. Calculate the sizes of a, b, c and d.

4. Calculate the size of x.
5. Calculate the size of x.
6. Calculate the size of x.
7. Calculate the sizes of a and C P.

including properties of triangles and quadrilaterals

1. Calculate the sizes of to .
2. RSTU is a trapezium. Calculate the sizes of and .
3. JKLM is a rhombus. Calculate the sizes of J L, 2 and 1.
4. ABCD is a parallelogram. Calculate the sizes of A B, A D, and D C.

1. Look at the drawing below. Name the items listed alongside.

(a) a pair of vertically opposite angles (b) a pair of corresponding angles (c) a pair of alternate interior angles (d) a pair of co-interior angles

2. In the diagram, AB// CD. Calculate the sizes of F G, , and . Give reasons for your answers.

 

3. In the diagram, OK = ON, KN// LM, KL// MN and L O = 160°.

Calculate the value of x. Give reasons for your answers.

 

Revision 232

• • Algebraic expressions 2 232
• • Algebraic equations 2 235
• • Construction of geometric figures 236
• • Geometry of 2D shapes 240
• • Geometry of straight lines 242

Assessment 244

Revision

Show all your steps in your working.

 

algebraic expressions 2

1. Simplify:

(a) x2 + x2

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(b) m + m \times m + m

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(c) 5ab – 7a2 – 2a2 + 11ba

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(d) (3ac2)(-4a2b)

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(e) (–4a2b3)3

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(f)

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(g)

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(h)

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(i) (2x + 3x)3

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(j) 3x2(4x3 - 5)

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(k) (4a - 7a)(a2 - 2a - 5)

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(l)

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(m)

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(n)

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(o)

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2. Simplify the following expressions:

(a) 3(a+ 2b) - 4(b - 2a)

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(b) 3 - 2(5x2 + 6x - 2)

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(c) 2x(x2 - x + 1) - 3(4 - x)

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(d) (2a + b - 4c) - (5a + b - c)

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(e) a{2a2[4 + 2(3a + 1)] - a}

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3. If a = 0, b = –2, and c = 3, determine the value of the following without using a calculator. Show all working:

(a) b2c

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(b) 2b – b(ab – 5bc)

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(c)

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4. If y = –2, find the value of 2y3 – 4y + 3

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algebraic equations 2

1. Solve the following equations:

(a) -x = -7

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(b) 2x = 24

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(c) 3x - 6 = 0

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(d) 2x + 5 = 3

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(e) 3(x - 4) = -3

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(f) 4(2x - 1) = 5(x - 2)

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2. Sello is x years old. Thlapo is 4 years older than Sello. The sum of their ages is 32.

(a) Write this information in an equation using xas the variable.

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(b) Solve the equation to find Thlapo's age.

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3. The length of a rectangle is (2x + 8) cm and the width is 2 cm. The area of the rectangle is 12 cm2.

(a) Write this information in an equation using x as the variable.

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(b) Solve the equation to determine the value of x.

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(c) How long is the rectangle?

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4. The area of a rectangle is (8x2 + 2x) cm2, and the length is 2x cm. Determine the width of the rectangle in terms of x, in its simplest form.

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construction of geometric figures

Do not erase any construction arcs in these questions.

1. (a) Construct D F = 56° with your ruler, pencil, and a protractor. Label the angle correctly.

 

(b) Bisect D F using only a compass, ruler, and pencil (no protractor).

2. Here is a rough sketch of a quadrilateral (NOT drawn to scale):

 

Construct the quadrilateral accurately and full size below.

 

3. Using only a compass, ruler and pencil, construct:

(a) A line through C perpendicular to AB

(b) A line through D perpendicular to AB

 

4. Construct and label the following triangles and quadrilaterals:

(a) Triangle ABC, where AB = 8 cm; BC = 5,5 cm and AC = 4,9 cm

 

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(b) Rhombus GHJK, where GH = 6 cm and = 50°

 

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5. Here is a rough sketch of triangle FGH (NOT drawn to scale):

 

Using a ruler, pencil, and protractor, construct and label the triangle accurately.

 

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6. Construct an angle of 120° without using a protractor.

 

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geometry of 2D shapes

1. True or false: all equilateral triangles, no matter what size they are, have angles that equal 60°.

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2. (a) In a triangle, two of the angles are 35° and 63°. Calculate the size of the third angle.

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(b) In a quadrilateral, one of the angles is a right angle, and another is 80°. If the remaining two angles are equal to each other, what is the size of each?

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3. If triangle MNP has = 40° and = 90°, what is the size of ?

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4. Write definitions of the triangles in the table below.

Equilateral triangle	Isosceles triangle	Right-angled triangle
A triangle which has all sides equal and so each angle is equal to 60°.	A triangle in which two of the sides are equal. The angles opposite the equal sides are also equal.	A triangle in which one angle is equal to 90°.

5. The following list gives the properties of three quadrilaterals, A, B and C.

(a) Give the special names of each of shapes A, B and C.

Quadrilateral A: The opposite sides are equal and parallel.

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Quadrilateral B: The adjacent sides are equal, while the opposite sides are not equal.

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Quadrilateral C: All of the angles are right angles.

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(b) What property must Quadrilateral A also have to make it a rhombus?

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(c) What property must Quadrilateral A also have to make it a rectangle?

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6. Determine the size of . Show all steps of your working and give reasons.

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x = 30°

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7. Determine the size of x. Give reasons.

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geometry of straight lines

1. Study the diagram alongside:

(a) Name an angle that is vertically opposite to E G.

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(b) Name an angle that is corresponding to E G.

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(c) Name an angle that is co-interior with E G.

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(d) Name an angle that is alternate to E G.

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2. Determine the size of x in each of the following diagrams. Show all steps of working and give reasons.

(a)

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x = 18°

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(b)

x + 90° = 110° [corr. ∠s; DF // GJ]

x = 110° - 90°

x = 20°

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(c)

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(d)

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x = 180° - 2(50°) [sum of ∠s of Δ = 180°]

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(e) Are line segments AB and DE parallel? Prove your answer.

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Assessment

In this section, the numbers in brackets at the end of a question indicate the number of marks the question is worth. Use this information to help you determine how much working is needed. The total number of marks allocated to the assessment is 75.

 

1. Simplify the following expressions:

(a) 5x2 – 6x2 + 10x2 (1)

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(b) 4(3x – 7) – 3(2 + x) (2)

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(c) (–2a2bc3)2 \div 4abcd (3)

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(d) (3)

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(e) (3)

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(f) 2[3x2 – (4 – x2)] – [9 + (4x)2] (3)

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2. Find the value of a if b = 3, c = –4 and d = 2:

(a) a = b + c \times d (2)

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(b) ab2 = 2c – d \div 2 (3)

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3. Solve the following equations:

(a) –7x = 56 (2)

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(b) 4(x + 3) = 16 (2)

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4. Sipho, Fundiswa and Ntosh are brothers. Sipho earns Rx per month; Fundiswa earns R1 000 more than Sipho per month, and Ntosh earns double what Sipho earns. If you add their salaries together you get a total of R27 000.

(a) Write this information in an equation using x. (2)

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(b) Solve the equation to find how much Fundiswa earns per month. (2)

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5. Construct the following figure using only a pencil, ruler and compass. Do not erase any construction arcs.

(a) An angle of 60° (2)

(b) The perpendicular bisector of line VW, where VW = 10 cm (3)

 

(c) Triangle KLM, where KL = 8,3 cm; LM = 5,9 cm and KM = 7 cm (4)

 

(d) Parallelogram EFGH, where E = 60°, EF = 4,2 cm and EH = 8 cm (4)

 

6. (a) What is/are the property/properties that make a rhombus different to a parallelogram? (1)

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(b) True or false: a rectangle is a special type of parallelogram. (1)

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7. Determine the size of x in each figure. Show all the necessary steps and give reasons.

(a)

(3)

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(b)

(4)

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(c)

(3)

x = 180° - 2(38°) [isos. Δ and sum of ∠s in Δ]

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8. Study the following diagram. Then answer the questions that follow:

 

(a) Write down the correct word to complete the sentence: x and y form a pair of

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angles. (1)

(b) Write down an equation that shows the relationship between angles x and y. (1)

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9. Determine the size of x, showing all necessary steps and giving reasons for all statements that use geometrical theorems:

(a)

(4)

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(b)

(5)

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x = 105° \div 3 = 35°

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(c)

(3)

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x = ED = 72° [corr. ∠s AB // EC]

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10. Consider the following diagram, in which it is given: D I = 30°, DE = EI, DF // IG, and GH = IH.

 

(a) Determine, with reasons, the size of . (6)

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(b) Which of the following statements is correct? Explain your answer. (2)

(i) \triangle}DEI is similar to \triangle}GHI

(ii) \triangle}DEI is congruent to \triangle}GHI

(iii) We cannot determine a relationship between \triangle}DEI and \triangle}GHI since there is not enough information given.

Statement

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is correct because

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